High School Mathematics-Physics SMILE Meeting
09 March 2004
Notes Prepared by Porter Johnson

Walter McDonald [CPS Substitute -- V A Technician]          Questions that Involve Sailing
Walter posed some questions that involved determining the direction of motion of a sailboat, when the directions of the wind velocity and the water current, as well as the orientation of the boat and location of the sails, were specified.  These questions appeared as exercises 877-980 (questions on p 303; answers on p 402) in the innovative book 1000 Play Thinks:  Games of Art, Science, and Mathematics by Ivan Moscovitch [at Amazon.com].  Roy Coleman mentioned this book in the Math-Physics SMILE class of 24 September 2002:  mp092402.html. Here is a brief paraphrased sample of questions and answers:

Question: Answer:
If you are sailing downwind in a 40 km/hr breeze, with your sail making a 90° angle with the keel of the boat, what is the  fastest speed you can achieve? Less than 40 km/hr. At that speed, the sails sag, as they would on a windless day.
If you are sailing downwind in a 40 km/hr breeze, with your sail making an angle of less than 90° with the keel of the boat, what is the fastest speed you can achieve? Less than before, since the sail catches less wind, and the force on the sails is not in the direction of motion of the boat.
If you are sailing crosswind in a 40 km/hr breeze, with your sail making an angle of less than 90° with the keel of the boat. Can you sail faster than in a tailwind? Yes, since the sail does not catch up with the wind, as in the first case.
How do you get a boat to move forward during a headwind? By tacking, or sailing on a zigzag course at an angle to the direction from which the headwind is coming, with the sails opened to catch the wind.

From now on, it will be smooth sailing for all of us.  Very good, Walter!

Fred Farnell [Lane Tech, Physics] Twinkling Shoes - Circuitry
At our last meeting [mp022404.html], Fred had dissected his daughter's twinkling tennis shoes in an effort to find out how they worked. He had found -- near the center of the sole -- a hard, transparent, plastic box, about 3 cm (square) and 1 cm thick, with wires connected to various LEDs imbedded in the shoe. But we ran out of time to figure out how it worked.

Now Fred held up a cardboard sheet (about 20 cm ´ 30 cm) on which he displayed to us the complete circuit, consisting of the LEDs connected to the box. He tapped the back of the cardboard against his hand (analogous to a shoe striking the ground), and sure enough! The LEDs twinkled on-and-off in some sort of sequential pattern, as they evidently had done in the shoe! "Inside the box," Fred told us, "you can identify a battery, a capacitor, a kind of switch, and transistor circuitry." Next, Fred brought a small (about 2.5 cm square, 6mm thick) permanent disk magnet near the box. Depending on where he placed the magnet, we watched the LEDs flash through a sequential pattern once, or to continually do so, or to light a single LED continuously! Fascinating! The switch appeared to be some sort of reed switch. Our speculations as to the basic mode of operation at the last meeting were only partially correct; it was more complex than most of us had assumed. But who would guess that such electronics would be found embedded in tennis shoes? From the patent number [US Patent Number 5,9969,479] on the box, Fred's colleague, Don Kanner, looked up the patent on the website of the United States Patent and Trademark Office: http://www.uspto.gov/patft/index.html. (Don has had experience in obtaining a patent a few years ago.) The following abstract describes this Light Flashing System, patented by Wai Kai Wong:

"A light-flashing system for flashing lights on and off and for generating a pattern of illumination for a plurality of lights in response to intermittent switch closures. The system includes a battery, a plurality of light-emitting elements, a plurality of transistors which enable the illumination of the light-emitting elements, a switch, a capacitor, and a pattern-generation circuit. The battery powers the light-emitting elements and the pattern-generation circuit. The switch intermittently clocks the pattern-generation circuit and enables the flow of current in certain of the transistors, allowing illumination of certain of the light-emitting elements in response to changes in inertial forces caused by movement of the flashing light system. The capacitor is connected in parallel to the battery such that the capacitor stores electrical charge when the switch is closed and continues to enable the flow of current through certain of the transistors after the switch is opened. The pattern-generation circuit then causes at least one, but not necessarily all, of the plurality of light-emitting elements to illuminate by enabling the flow of current through certain of the transistors. As the switch intermittently opens and closes, the pattern-generation circuit is clocked through various states, and the outputs of the pattern-generation circuit enable the flow of current through certain of the transistors, allowing illumination of at least one, but not necessarily all, of the light-emitting elements in a pattern."
The patent was given for a "shoe" system. Could one get a new patent for a flashing system for boots??  If so, those boots aren't just for walking, are they? (you can search for patent information through http://www.freepatentsonline.com/)

You really held our feet to the fire for this one! Great job, Fred.

Bill Shanks [Joliet Central, Physics -- retired]           Pop Can Electroscope: Construction and Operation
Holding a home-made apparatus up for us to see, Bill asked, "Does anybody know what this is?" Somebody guessed, "An electroscope?" In response, Bill rubbed a small, inflated balloon on his head and held it near the apparatus. We saw a small, gold-colored, metallic strip pivot back-and-forth as he moved the balloon toward-and-away from the apparatus. "We're all going to to build an electroscope like this, to take home," said Bill. Then he gave us the following items:

We turned the Styrofoam® cup upside down on a level surface, where it served as an insulating base.  The tape was placed along the cylindrical surface of the can, in rough alignment with the bottom of the tab opening on its end. The tape was folded back on itself, with its adhesive holding the horizontal can to the top end of the inverted cup.  A strip of foil  (about 1 cm ´ 4 cm) was cut, and an end of the narrow side was wrapped partially around the toothpick, in order to shape the strip like a hook.  The strip was then hooked onto the top end of the tap, so that it could pivot.  It took some patience and careful attention to get the right configuration. We then blew up the balloon, knotted it shut, and rubbed it against our hair or clothing to build up a static charge on it.  When the charged balloon was brought close to one end of the can, the strip on the other end moved away from the metallic surface of the can.  Like charges repel! Using Bill's electroscope as a model, it was fairly easy for each us to make his/her own.  We used our apparatus to explore and observe electrostatic interactions, including the following: Bill made a sketch of the pop can electroscope on the blackboard. On it, he placed small, round (about 2 cm diameter) red magnets to represent positive charges, and gray magnets to represent equal negative charges. [These magnets are available at office supply houses] Each red magnet was paired with a gray one to represent a net charge of "0". Then he drew a picture of the balloon with positive (red) charge on it, close to one end of the electroscope. To illustrate the resulting redistribution of charge on the pop can, Bill moved the negative (gray) charges on the pop can closer to the balloon (unlike charges attract!), and the positive (red) charges away from it (like charges repel!), resulting in a positive charge on the other end of the pop can and on the pivoting, foil strip. Bill pointed out that he gets his students to the blackboard to show electrostatic interactions using such magnets. 

For related information on Home-made Electroscopes see http://www.schoolinyourhome.com/science/electroscope.htm.

Beautiful phenomenological physics, Bill!

Fred Schaal [Lane Tech, HS Mathematics]           When is there more crust than "pie"?
Fred carefully drew a full circle on the blackboard, and marked the center of the circle, as well as two points on its circumference. He drew radial lines (r) from the center of the circle to these two points A, B --- enclosing a sector, or a pie-shaped slice of circle. Then he drew a straight line (chord) connecting A and B.  The area between the chord and the arc of the circle represents the crust, or outer region of the slice  Here is a rough diagram.
The slice angle between the two radii is q.  In terms of area, how much crust is there, and how much pie, for a given radius r and slice angle q?  Fred explained that the slice area  is Aslice =½ r2 q, with the slice angle q measured in radians, since the slice area is proportional to q, and for q = 2p (full circle) we get an area of pr2.  Fred then measured the radius r of his circle and the slice angle q, and calculated the slice area Aslice.

How do you determine the area Atri of the central  "edible" portion of the pie -- the slice minus the crust? It is the triangular region of base b and altitude h; so that  Atri = ½ b h.  Fred measured the chord length b and triangle altitude h directly on the diagram, and calculated Atri.  Alternatively, we can compute them using h = r sin q/2 and b = 2 r cos q/2, so that Atri =  r2 sin q/2  cos q/2 = ½ r2 sin q. Therefore, the fraction of the slice area that is crust area is given by

Acrust / Aslice = 1 - Atri / Aslice = 1 - sin q / q

This result, depends only on the slice angle q, and is independent of the pie radius.  Why? This table gives its value for various ways of slicing the pie into equal pieces:.

Number of Slices q: degrees   q: radians   sin q / q Acrust / Aslice  
12 30° 0.523 0.956 0.044
8 45° 0.785 0.901 0.099
6 60° 1.047 0.827 0.173
4 90° 1.571 0.637 0.373
3 120° 2.094 0.414 0.586
2 180° 3.142 0.000 1.000

Thanks for sharing the pie for us --- figuratively speaking!  Good, Fred!

We ran out of time before these three presentations could be made:

  1. Bill Colson: Euler's Disk.
  2. Bill Blunk: Kelvin Water Droplet Electrostatics
  3. Ann Brandon: Current without Batteries??
 They will be scheduled at the beginning of our next class, Tuesday 23 March 2004.  See you there!

Notes prepared by Porter Johnson